All Questions
Tagged with mathematica complex-analysis
26
questions
1
vote
1
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95
views
Fourier transform of incomplete gamma function
Ultimately I am interested in the Fourier transform of
$$
e^{-i\zeta}(-i\zeta)^{-2\epsilon}\Gamma(2\epsilon,-i\zeta)
$$
in a series expansion around $\epsilon=0$, so to first order in
$$
\lim_{\...
- 133
0
votes
2
answers
195
views
what are the branch points and branches of $g(z)=(z+ \sqrt{z})^{1/3}$?
And what if we for example shifted one of the roots, eg $f(z)=(z+ \sqrt{z-3})^{1/3}$?
I already asked a more extensive version of this question here Branch cut/ points for square roots inside cubic ...
- 67
0
votes
0
answers
92
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Branch cut/ points for square roots inside cubic roots- incorrect branching by mathematica or my mistake?
There's a lot of great information here about understanding the branch cuts and branch points of functions of the form ( for example ) $(z^3+1)^{1/2}$, sums of simple roots and products thereof.
...
- 67
1
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1
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327
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Closed form expression for an integral
Let $\psi_q(z)$ be the q-DiGamma function defined for a complex variable $z$ with $\Re(z)>0$ as $$\psi_q(z)=\frac{1}{\Gamma_q(z)}\frac{\partial}{\partial z} (\Gamma_q(z))$$
where $\Gamma_q(z)$ is ...
- 910
4
votes
1
answer
145
views
Closed form expression for $\psi_{e^{\pi}}^{(3)}(1-i)$
Let $\psi_q(z)$ be the q-DiGamma function defined for a real variable $\Re(z)>0$ as $$\psi_q(z)=\frac{1}{\Gamma_q(z)}\frac{\partial}{\partial z} (\Gamma_q(z))$$
where $\Gamma_q(z)$ is the q-Gamma ...
- 910
0
votes
1
answer
132
views
Closed form expression for $\psi_{e^{\pi}}^{(3)}(1)$
Let $\psi_q(x)$ be the q-DiGamma function defined for a real variable $x>0$ as $$\psi_q(x)=\frac{1}{\Gamma_q(x)}\frac{\partial}{\partial x} (\Gamma_q(x))$$
where $\Gamma_q(x)$ is the q-Gamma ...
- 910
3
votes
0
answers
168
views
Elliptic integrals of complex argument and parameter
Mathematica has the annoying habit to provide solutions involving incomplete elliptic integrals of the first kind $E(z|m)$, second kind $F(z|m)$ and third kind $\Pi(...
- 69
3
votes
1
answer
388
views
Hermite polynomials for non-integer degree
I have solved an eigenvalue problem using Mathematica and the answer is in terms of Hermite polynomials.
Now, for integer degrees $H_n(z)$, I can find a nice definition. However, in the solution to ...
- 349
3
votes
1
answer
112
views
Cauchy's theorem and mathematica disagree? Integral involving branch points.
Consider the following integral:
$$\int_{-\infty}^{\infty} \frac{dx}{\sqrt{x^2-2i\epsilon x -1}(x^2+1)}$$
where $\epsilon$ is an infinitesimal positive number. In the complex $x$-plane, the integrand ...
- 3,968
1
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0
answers
83
views
Conditions on the coefficients that the roots of a polynomial be less that or equal to unity in absolute value
Consider the polynomial $$f(x)=p_0x^n+p_1x^{n-1}+...+p_{n-1}x+p_n,~p_i \in \mathbb C.$$
Particularly, in the case of absolute stability of a multi-step numerical method, how can we find out the ...
- 2,184
6
votes
1
answer
219
views
Why do Mathematica and Wolfram|Alpha say $\Gamma(-\infty)=0$?
According to Mathematica and Wolfram|Alpha, $\lim_{x\to -\infty}\Gamma(x)$ is equal to zero. See e.g https://www.wolframalpha.com/input/?i=gamma+function (at the bottom of the page), or try ...
- 71
0
votes
1
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295
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How can I evaluate this complex integral equation on Wolfram?
I need to evaluate the complex line integrals in the following equation:
$$g(z)=\frac{\int_0^z\zeta^{-5/6}(\zeta-1)^{-1/2}d\zeta}{\int_0^1\zeta^{-5/6}(\zeta-1)^{-1/2}d\zeta}.$$
Can someone advise me ...
- 477
0
votes
0
answers
37
views
Error in Taylor series using Mathematica
How many terms should be used in taylor series expansion of the function f(z) = e^z around z = 0 for a specific value of z = 30 + 30 i to get an error of less than 0.05 using Mathematica?
...
1
vote
1
answer
88
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Proving $ \sum^{\infty}_{k=1}\frac{\textrm{sin}(\frac{k\pi}{2})\textrm{cos}(k\delta)}{k}=1/4({\pi}+gd(i\delta)+gd(-i\delta))$
In Mathematica, we can show that
\begin{equation}\label{gd}
\sum^{\infty}_{k=1}\frac{\textrm{sin}(\frac{k\pi}{2})\textrm{cos}(kx)}{k}=\frac{1}{4}\big({\pi}+\textrm{gd}(ix)+\textrm{gd}(-ix)\big),
\end{...
- 343
0
votes
0
answers
155
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Can any Gaussian integral with complex limits be written as a (complex) error function?
Can the integral,
$$I = \int^{-x}_{-\infty} e^{-at^2}\ \mathrm dt,\ a \in \mathbb{C},\ Re(a) > 0,$$
be written as an error function?
I tried, by substitution,
$$\int^{-x}_{-\infty} e^{-at^...
- 13